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On the consistency of Sobol indices with respect to
stochastic ordering of model parameters.
Areski Cousin, Alexandre Janon, Véronique Maume-Deschamps, Ibrahima
Niang
To cite this version:
RESPECT TO STOCHASTIC ORDERING OF MODEL PARAMETERS.
A. COUSIN, A. JANON, V. MAUME-DESCHAMPS, AND I. NIANG
Abstract. In the past decade, Sobol’s variance decomposition have been used as a tool - among others - in risk management ([3, 14]). We show some links between global sensitivity analysis and stochastic ordering theories. This gives an argument in favor of using Sobol’s indices in uncertainty quantification, as one indicator among others.
Introduction
Many models encountered in applied sciences involve input parameters which are often not precisely known. Sobol indices are used so as to assess the sensibility of a model output as a function of its input parameters. In other words, they quantify the impact of inputs’ uncertainty on an output. Sobol indices are widely used for example in hydrology (see [21, 15]). Re-cently ([3, 14, 16]), this global sensitivity analysis has been used as a risk management tool, amongst other indicators ([2]). In this work, our goal is to prove that Sobol indices behave coherently with respect to stochastic order-ing theory. This question arises naturally: under some reasonable conditions on the output, uncertainty quantifiers should increase if the uncertainty on the input increases in some way. Thus, our problematic is to find for which kind of stochastic orders and under which sufficient conditions on the output function, the Sobol indices behave coherently.
Roughly speaking, given two random variables X and Y , the X-Sobol inde-ces on Y is given by
SX =
Var(E(Y | X)) Var(Y ) .
It is a statistical indicator of the relative impact of X on the variability of Y . If we study the impact of several independent variables X1, . . . , Xk on
Y , then the Sobol indices may be used to provide a hierarchization of the Xi’s with respect to their impact on Y . It is an interesting alternative to
regression coefficient, which may be hardly interpreted if the relationship between Y and the Xi’s is far from linear.
In our context, X = (X1, . . . , Xk) is a random vector of Rk, with the
Xi’s being independent. We are interested in the variance and then the
Sobol indices of an output function f . We shall assume properties such as convexity and/or monotonicity of the function f . One of our main result is that Sobol indices have a behavior which is compatible with respect to the excess wealth order / or the dispersive order (it depends on the convexity properties of f , see Theorems 2.1 and 2.3). The fact that Sobol index are
in accordance with the excess wealth or with the dispersive order confirms that it could be used to quantify some uncertainty, even if, depending on the purpose, moment-independent approaches should be prefered to vari-ance decomposition (see [1]). Neverthelss, as we shall see in the examples, the ordering of the Sobol indices heavily depends on the law of the param-eters, so that one has to be careful on the conclusions.
To simplify notations, if i ∈ {1, . . . k}, we shall write X−i for the random
vector (X1, . . . , Xi−1, Xi+1, . . . , Xk), and if α ⊂ {1, . . . k}, we shall write
Xα for the random vector (Xi, i ∈ α) and X−α for the random vector
(Xi, i6∈ α).
The paper is organized as follows. In Section 1, we study the impact of stochastic orders on the variance. In Section 2, we state our may results concerning the accordance of Sobol indices with respect to the dispersive order. Finally, in Section 3 we provide some examples of illustrations. In Section 4, we give some concluding remarks.
1. Impact of the stochastic orders on the variance Let us recall some particular notions of ordering on random variables / vectors. We refer to [13, 17] or [7] for a detailed review on stochastic orders, their relationships and properties.
1.1. Stochastic orders. We shall be mainly interested in the stochastic order, the convex order, the dispersive order, the excess wealth order, the ∗ order and the Lorenz order. For a random variable X, FX denotes its
distribution function, and FX−1the generalized inverse of FX (or the quantile
function). The survival function is FX = 1− FX.
Definition 1.1. Let X and Y be two random variables, we say that (1) X is smaller than Y for the standard stochastic order (X≤st Y ) if
and only if, for any bounded and non decreasing function f ,
E(f (X))≤ E(f(Y )).
(2) X is smaller than Y for the convex order (X ≤cx Y ) if and only if, for any bounded convex function f ,
E(f (X))≤ E(f(Y )).
(3) If X and Y have finite means, then X is smaller than Y for the dilatation order (X ≤dil Y ) if and only if
(X− E(X)) ≤cx (Y − E(Y )).
(4) X is smaller than Y for the dispersive order (X ≤disp Y ) if and only if FY−1− FX−1 is non decreasing.
(5) If X and Y have finite means, then X is smaller than Y for the excess wealth order (X≤ew Y ) if and only if for all p ∈]0, 1[,
(6) If X and Y are non negative, X is smaller than Y for the star order (X ≤* Y ) if and only if
FY−1
FX−1 is non decreasing.
(7) If X and Y are non negative with finite mean, X is smaller than Y for the Lorenz (X≤Lorenz Y ) if and only if
X E(X) ≤cx
Y E(Y ).
Remark 1.2. The st and cx orders may be defined in the same way for random vectors.
1.2. Some relationships between variance and stochastic orders. It is well known that the stochastic order and the convex order are not location-free and may not be compatible with the variance. The dispersive and excess wealth orders are location-free and in accordance with the variance. Below, we give some conditions implying some accordance of the stochastic order or the convex order with respect to the variance.
Proposition 1.3. Let i ∈ {1, . . . , k}. If X∗
i is a random variable, we
shall write Xi∗ for the random vector of Rk: Xi∗ = (X
1, . . . , Xi−1, Xi∗,
Xi+1, . . . , Xk). Let Xi∗ be a random variable, independent of X. The
fol-lowing holds: (1) If X∗
i ≤st Xi then Xi∗ ≤st X. In particular, if f is non decreasing
with respect to its i-th component and E(f (Xi∗)) = E(f (X)) then Var(f (Xi∗))≤ Var(f(X)).
(2) If Xi∗ ≤cx Xi then Xi∗ ≤cx X. In particular, if f is convex
with respect to its i-th component and E(f (Xi∗)) = E(f (X)) then Var(f (Xi∗))≤ Var(f(X)).
Proof. The result follows from the definitions and the fact that for any
func-tion f : Rk −→ R,
E(f (X)) = E E(f (X1, . . . , Xk)|X−i) .
In what follows, we will consider the excess wealth order, for which we can prove the ordering of Sobol’s indices. Let us remark that the dispersive order implies the excess wealth order. Natural examples of random variables ordered with respect to the dispersive order will be recalled in Section 3. The following results proved in [17], show that the excess wealth order (and thus the dispersive order) is in accordance with the variance.
Proposition 1.4. [17] Let X and Y be two random variables with finite means.
(2) If X and Y are non negative and X ≤* Y then X ≤Lorenz Y and then Var(X) E(X)2 ≤ Var(Y ) E(Y )2 .
(3) If X and Y are non negative then X≤* Y if and only if log X ≤disp
log Y .
(4) If X ≤disp Y and X ≤st Y then for all non decreasing convex or non increasing concave function f , f (X)≤disp f(Y ).
(5) If X and Y are continuous random variables with supports bounded from below by (resp.) ℓ∗ and ℓ, X ≤ew Y and −∞ < ℓ∗ ≤ ℓ, then
for all non decreasing and convex function f , for which f (X) and
f (Y ) have finite means, we have f (X)≤ew f(Y ). [Theorem 4.2 in
[18]]
Remark 1.5. Result (5) above has been incorrectly stated in [11] and [17], where the hypothesis on the left-end points of the supports was missing. This hypothesis is indeed required, as shown by the following example.
Consider X which follows a uniform law on [1, 1.9] and Y which follows a uniform law on [0, 1]. Then X ≤ew Y . Let f = exp which is a convex and
increasing function. We have that Var(f (X)) ∼ 1.32 and Var(f(Y )) ∼ 0.24 so that f (X) cannot be less than f (Y ) for the ew order (see (1) of Proposition 1.4). The correct statement and proof of (5) above may be found in [18], as well as an example showing that the left-end points of the support have also to be finite.
Below, we give two simple counter-examples that show that the hypothesis on f above are necessary to get the inequality on the variance.
Example 1.6. Let X have uniform law on [0, 1] and Y have uniform law on [0, 10]. We consider the function f such that f (t) = t for t ∈ [0, 1] and f (t) = 1 for t ≥ 1. f is a non decreasing function and X ≤st Y . But, Varf (X) > Varf (Y ).
Example 1.7. Let X be such that P(X = 0) = 1920 and P(X = 1) = 201 and Y be such that P(Y = 0) = 12 and P(Y = 10) = 12. Consider, any function f such that f (0) = 0, f (1) = 10 and f (10) = 1. Then, we have
E(f (X)) = E(f (Y )), X ≤cxY but Var(f (X)) > Var(f (Y )).
2. Impact on Sobol indices
Sobol indices can be used as a tool to quantify the impact of input pa-rameters on the output. They are more accurate than the variance in order to identify the input variables that have the most important impact on the output. Our goal is to explore how an increase of riskness (in the sense of stochastic orders) of the input parameters may have an impact on the out-put. We begin by recalling definitions on Sobol indices. We refer to [5, 19] or [10] for more details on this subject.
2.1. Some facts on Sobol indices. As before, we consider one output Y = f (X1, . . . , Xk) with X1, . . . , Xkindependent random variables. In what
shall denote µXα the law of the random vector Xα. For α⊂ {1, . . . , k}, |α|
denotes the length of α, i.e. its number of elements. The function f can be decomposed into
(2.1) f (X1, . . . , Xk) = X α⊂{1,...,k} fα(Xα), with (1) f∅ = E(f (X)), (2) Z fαdµXi = 0 if i∈ α, (3) Z fα· fβdµX = 0 if α6= β.
The functions fα are defined inductively:
f∅= E(f (X)),
for i∈ {1, . . . , k}
(2.2) fi(Xi) = E(f (X)| Xi)− f∅=
Z
f dµX−i− f∅.
If the fβ have been defined for|β| < n, let α ⊂ {1, . . . , k} with |α| = n then,
fα(Xα) = Z f dµX−α− X β(α fβ(Xβ).
With these notations, we have that:
Var(Y ) = Var(f (X)) = X α⊂{1,...,k} Var(fα(Xα)) = X α⊂{1,...,k} E(fα(Xα)2).
This decomposition of variance is often called Hoeffding decomposition ([20]). The impact of Xi on Y = f (X) may be measured by the Sobol index:
(2.3) Si =
Var(E(f (X) | Xi))
Var(Y ) =
E(fi(Xi)2)
Var(Y ) .
There are also interactions between the variables X1, . . . , Xk, they are
iden-tified by the fα, with |α| ≥ 2. The total Sobol indices take into account the
impact of the interactions:
(2.4) STi = X i∈α⊂{1,...,k} Var(fα(Xα)) Var(Y ) = X i∈α⊂{1,...,k} E(fα(Xα)2) Var(Y ) .
2.2. Relationship with stochastic orders when there is no interac-tions. In this section, we assume that there is no interactions between the Xi’s, that is, f (X) can be expressed in the following additive form:
(2.5) f (X1, . . . , Xk) = k
X
j=1
where g1, . . . , gk are real-valued functions and K ∈ R. It is straitghforward
to prove that, in that case, decomposition (2.1) reduces to
(2.6) f (X) =
k
X
i=1
fi(Xi) + f∅,
so that, for any i = 1, . . . , k, the “individual” Sobol index defined by (2.3) coincides with the total Sobol index defined by (2.4).
As in the previous section, Xi∗ denotes another variable that will be com-pared to Xi. We shall assume Xi∗≤ew Xi and study the impact of replacing
Xi by Xi∗ on Sobol indices. We assume that Xi∗ is independent of X−i and
we denote by X∗ = (X1, . . . , Xi−1, Xi∗, Xi+1, . . . , Xk) the vector X where
the i-th component has been replaced by X∗
i and by
Si∗= Var(E(f (X
∗)| X∗ i)))
Var(f (X∗))
the i-th Sobol index associated with f (X∗). Because we shall use the excess wealth order, we assume that Xi and Xi∗ have finite means, this hypothesis
may be relaxed by considering random variables ordered with respect to the dispersive order.
Theorem 2.1. We assume that there is no interactions, i.e. (2.5) is sat-isfied. Let Xi∗ be a random variable independent of X−i and assume that
X∗
i ≤ew Xi and −∞ < ℓ∗ ≤ ℓ, where ℓ∗ and ℓ are the left-end points of
the support of Xi∗ and Xi. If gi is a non decreasing convex function, then
S∗i ≤ Si and Sj∗ ≥ Sj for j 6= i.
The proof of Theorem 2.1 makes use of Proposition 1.4.
Proof. As gi is a non decreasing convex function, Proposition 1.4 implies
gi(Xi∗) ≤ew gi(Xi) and Var(gi(Xi∗)) ≤ Var(gi(Xi)). Now, the Hoeffding’s
decomposition of f (X) can be expressed as (2.5) where f∅ = E[f (X)] and
fj(Xj) = gj(Xj)− E[gj(Xj)] and the Hoeffding’s decomposition of f (X∗)
writes
f (X∗) =X
j6=i
fj(Xj) + fi∗(Xi∗) + f∅∗
where f∅∗ = E[f (X∗)] and fi∗(Xi∗) = gi(Xi∗)− E[gi(Xi∗)].
Then, from (2.3), the i-th Sobol indices of f (X) and f (X∗) are such that
and S∗i = Var(gi(Xi∗)) X j6=i Var(gj(Xj)) + Var(gi(Xi∗)) (2.8) = 1 + X j6=i Var(gj(Xj)) Var(gi(Xi∗)) −1 . (2.9)
We have already noticed that Var(gi(Xi∗)) ≤ Var(gi(Xi)) and thus we
conclude that S∗
i ≤ Si. The result for j 6= i follows from the fact that
Var(gi(Xi∗))≤ Var(gi(Xi)) and
Sj = Var(gj(Xj)) X j6=i Var(gj(Xj)) + Var(gi(Xi)) and S∗j = X Var(gj(Xj)) j6=i Var(gj(Xj)) + Var(gi(Xi∗)) . Remark 2.2. Note that, from Proposition 1.4, the previous result also holds for any non decreasing convex or non increasing concave function gi as soon
as X∗
i ≤disp Xi and Xi∗ ≤st Xi. In addition, it is shown in [17] that if
Xi∗ ≤disp Xi with common and finite left end points of their support (i.e.,
ℓ∗ = ℓ) then Xi∗≤st Xi.
2.3. Relationship with stochastic orders when there are interac-tions. In the case where there are interactions, we have to consider the total Sobol indices as defined by (2.4). We will first show that the i-th total Sobol indices are ordered if Xi∗ ≤disp Xi and Xi∗ ≤st Xi, provided that
the function f is a product of functions of one variable whose log is non decreasing and convex. Then we consider some extensions of that case. Theorem 2.3. We assume that f writes:
(2.10) f (X1, . . . , Xk) = g1(X1)× · · · × gk(Xk) + K
where K ∈ R and gj, j = 1, . . . , k are real-valued functions. Let Xi∗ be
a random variable independent of X−i and assume that X∗
i ≤disp Xi and
Xi∗ ≤st Xi. If log gi is a non decreasing convex or a non increasing concave
function, then ST∗i ≤ STi and S ∗
Tj ≥ STj, for j 6= i.
Proof. Without loss of generality, we may assume that K = 0. With the
hypothesis of Theorem 2.3, the decomposition (2.1) satisfies: for all j = 1, . . . , k,
fj(Xj) = (gj(Xj)− E(gj(Xj))
Y
ℓ6=j
E(gℓ(Xℓ)),
and following, e.g. [12], for α⊂ {1, . . . , k}, we have (2.11) fα(Xα) =
X
β⊂α
The form of f then gives: fα(Xα) = X β⊂α (−1)|α|−|β|Y j∈β gj(Xj) Y j6∈β E(gj(Xj)) = Y j6∈α E(gj(Xj)) Y j∈α (gj(Xj)− E(gj(Xj))) . We write fTi = X α∋i fα Then, fTi = X α∋i Y j6∈α E(gj(Xj)) Y j∈α (gj(Xj)− E(gj(Xj))) = (gi(Xi)− E(gi(Xi)) X γ⊂{1,...,k}\{i} Y j6∈γ E(gj(Xj)) Y j∈γ (gj(Xj)− E(gj(Xj))) . Now, X γ⊂{1,...,k}\{i} Y j6∈γ E(gj(Xj)) Y j∈γ (gj(Xj)− E(gj(Xj))) = Y j∈{1,...,k}\{i} (gj(Xj)− E(gj(Xj)) + E(gj(Xj))) = Y j∈{1,...,k}\{i} gj(Xj). So that, finally, (2.12) fTi(X) = (gi(Xi)− E(gi(Xi)) Y j6=i gj(Xj).
We denote by fα∗ the functions involved in the Sobol decomposition of f (X∗) and by f∗
Ti the sum of the f ∗
α’s over the α for which i∈ α. Then,
fT∗i(X∗) = (gi(Xi∗)− E(gi(Xi∗)) Y j6=i gj(Xj), and STi = Var(fTi(X)) Var(f (X)) and S ∗ Ti = Var(fT∗ i(X ∗)) Var(f (X∗)) .
As in the proof of Theorem 2.1, this may be rewritten as
We have Var(fTi(X)) = Var(gi(Xi)) Y j6=i E(gj(Xj)2) and Var(fT∗i(X∗)) = Var(gi(Xi∗)) Y j6=i E(gj(Xj)2). Also, if i6∈ α,
Varfα(Xα) = E(gi(Xi))2Var
Y j6=i j6∈α E(gj(Xj)) Y j∈α (gj(Xj)− E(gj(Xj))) and
Varfα∗(Xα) = E(gi(Xi∗))2Var
Y j6=i j6∈α E(gj(Xj)) Y j∈α (gj(Xj)− E(gj(Xj))) .
So, the result follows from (2.13) if Var(gi(Xi∗))
E(gi(Xi∗))2
≤Var(gE(g i(Xi))
i(Xi))2
.
We have (see Proposition 1.4)
log gi(Xi∗)≤disp log gi(Xi)⇐⇒ gi(Xi∗)≤∗gi(Xi)
so that gi(Xi∗)≤Lorenz gi(Xi) =⇒ Var(gi(Xi∗)) E(gi(Xi∗))2 ≤ Var(gi(Xi)) E(gi(Xi))2 and thus ST∗ i ≤ STi. When j 6= i, we have: Var(fTj(X)) = E(gi(Xi) 2)Var(g j(Xj)) Y p6∈{i,j} E(gp(Xp)2) and Var(f (X)) = X α⊂{1,...,k},α6=∅ Var(fα(Xα)) = E(g2i(Xi)) X α⊂{1,...,k}\{i},α6=∅ Y p6∈α E(gp(Xp))2 Y p∈α Var(gα(Xα)) +Var(gi(Xi)) Y p6=i E(gp(Xp))2. So that ST∗ j ≥ STj if E(gi(Xi∗)2) E(gi(Xi∗))2 ≤ E(gi(Xi) 2) E(gi(Xi))2
which holds as above because gi(Xi∗)≤Lorenz gi(Xi).
The conditions on stochastic ordering between Xi and Xi∗, and on the
Example 2.4. To see the necessarity of the stochastic ordering, one can consider:
f (X1, X2, X3) = exp(exp(X1)) exp(X2) exp(X3),
with X1, X2 and X3 uniform on [0, 1]. If X1∗ is uniform on [1, 1.9], then
X1∗ ≤disp X1 but X1 ≤st X1∗ and it can be easily checked that ST∗1 > ST1
(ST∗1 ≈ 0.90 and ST1 ≈ 0.65).
Example 2.5. The log-convexity of gi is also necessary. Indeed, take:
f (X1, X2, X3) = g1(X1)X2X3 where g1(x) = 0 if x < 0.45, x/10 if 0.45≤ x ≤ 0.5, x else.
and X2, X3 uniform on [2, 3], X1 uniform on [0, 1], X1∗ uniform on [0, 0.5]
so that X∗
1 ≤disp X1, X1∗ ≤st X1 but g1 is not log-convex. In that case, we
have S∗
T1 > ST1 (S ∗
T1 ≈ 0.99 and ST1 ≈ 0.97).
Now, we turn to the case where f writes as a sum of product of convex and non decreasing functions of one variable, that is, there are: a finite set A and convex and non decreasing functions ga
i, i∈ {1, . . . , k}, a ∈ A, such
that
(2.14) f (X) =X
a∈A
g1a(X1)× · · · × gak(Xk).
Proposition 2.6. Assume that f satisfies (2.14). Then, for any i∈ {1, . . . , k},
fi(Xi) = X a∈A (gai(Xi)− E(gia(Xi)) Y j6=i E(gja(Xj)) (2.15) fTi(X) = X a∈A (gai(Xi)− E(gia(Xi))) Y j6=i gja(Xj) (2.16) Var(fTi) = X a,b∈A Cov(gai(Xi), gib(Xi)) Y j6=i E(gja(Xj)gbj(Xj)). (2.17)
Proof. The proof uses in a straightforward way the computations done in
the proof of Theorem 2.3.
We deduce the two following extensions of Theorem 2.3.
Proposition 2.7. Let {Ia}a∈A be a partition of{1, . . . , k} and assume that
f (X) =X
a∈A
Y
j∈Ia
gj(Xj)
where the gj’s are real-valued functions. Let Xi∗ be a random variable
inde-pendent of X and assume that X∗
i ≤disp Xi and Xi∗ ≤st Xi. If log gi is a
non decreasing convex function or a non increasing concave function, then
S∗Ti ≤ STi and S ∗
Proof. The proof follows that of Theorem 2.3 in a straighforward way
be-cause the Iaare disjoints.
Proposition 2.8. Let f (X) = ϕ1(Xi)
Y
j6=i
gj(Xj) + ϕ2(Xi) with log ϕ1 and
log ϕ2 non decreasing and convex. If
• X∗
i is independent of X, Xi∗ ≤disp Xi and Xi∗ ≤st Xi.
• Var(ϕ2(X ∗ i)) E(ϕ1(Xi∗))2 ≤ Var(ϕ2(Xi)) E(ϕ1(Xi))2 and Cov(ϕ1(X ∗ i), ϕ2(Xi∗)) E(ϕ1(Xi∗))2 ≤ Cov(ϕ1(Xi), ϕ2(Xi)) E(ϕ1(Xi))2 . Then ST∗ i ≤ STi.
Proof. Proposition 2.6 gives
Var(fTi(X)) = Var(ϕ1(Xi)) Y j6=i E(gj(Xj))2+ Var(ϕ2(Xi)) +Cov(ϕ1(Xi), ϕ2(Xi)) Y j6=i E(gj(Xj)) and if i 6∈ α, fα(Xα) = E(ϕ1(Xi)) Y j6∈α E(gj(Xj))× Y j∈α (gj(Xj)− E(gj(Xj))), Var(fα(Xα)) = E(ϕ1(Xi))2 Y j6∈α E(gj(Xj))2×Var Y j∈α (gj(Xj)− E(gj(Xj))) .
We use once more (2.13) to get that ST∗i ≤ STi if and only if
Var(ϕ1(Xi∗))
Y
j6=i
E(gj(Xj))2+ Var(ϕ2(Xi∗)) + Cov(ϕ1(Xi∗), ϕ2(Xi∗))
Y j6=i E(gj(Xj)) E(ϕ1(X∗ i))2 X α6∋i Y j6∈α E(gj(Xj))2× Var Y j∈α (gj(Xj)− E(gj(Xj))) ≤ Var(ϕ1(Xi)) Y j6=i
E(gj(Xj))2+ Var(ϕ2(Xi)) + Cov(ϕ1(Xi), ϕ2(Xi))
Y j6=i E(gj(Xj)) E(ϕ1(X∗ i))2 X α6∋i Y j6∈α E(gj(Xj))2× Var Y j∈α (gj(Xj)− E(gj(Xj))) .
With our hypothesis, we have that Var(ϕ1(Xi∗)) E(ϕ1(Xi∗))2 ≤Var(ϕ1(Xi)) E(ϕ1(Xi))2 , Var(ϕ2(Xi∗)) E(ϕ1(Xi∗))2 ≤ Var(ϕE(ϕ 2(Xi)) 1(Xi))2 and Cov(ϕ1(Xi∗), ϕ2(Xi∗)) E(ϕ1(Xi∗))2 ≤ Cov(ϕ1(Xi), ϕ2(Xi)) E(ϕ1(Xi))2 .
This leads to the announced result.
Example 2.9. Let Xi ∼ U([1.5, 3.5]), Xi∗ ∼ U([0, 1.8]), ϕ1(x) = exp(x2)
and ϕ2(x) = exp(x), gj(x) = 1, j6= i. Then one can show that Xi∗≤disp Xi
and Xi∗ ≤st Xi. However, Var(ϕ2(Xi∗)) E(ϕ1(Xi∗))2 ∼ 0.08 Var(ϕ2(Xi)) E(ϕ1(Xi))2 ∼ 10 −7 Cov(ϕ1(Xi∗), ϕ2(Xi∗)) E(ϕ1(Xi∗))2 ∼ 0.3 Cov(ϕ1(Xi), ϕ2(Xi)) E(ϕ1(Xi))2 ∼ 10 −3.
and, from the proof of Proposition 2.8, S∗
Ti > STi.
The following result derived in [8] is mentioned here as a related result on excess wealth orders, even if it is not sufficient to obtain a more general version of Proposition 2.8.
Proposition 2.10. [Corollary 3.2 in [8]] Let X and Y be two finite means random variables with supports bounded from below by ℓX and ℓY
respec-tively. If X ≤ew Y and ℓX ≤ ℓY then for all non decreasing and convex
functions h1, h2 for which hi(X) and hi(Y ) i = 1, 2 have order two moments,
(2.18) Cov(h1(X), h2(X))≤ Cov(h1(Y ), h2(Y )).
3. Examples
In this section, we illustrate the previous results on some classical financial risk models. All considered models are associated with a set of parameters. In a context where these parameters are not known with certainty (due to es-timation error for instance), the global sensibility analysis is useful to assess which (uncertain) input parameters mostly contribute to the uncertainty of model output and in turns, which parameters have to be estimated with caution. In most of our examples, we will consider truncated distribution functions (ordered with respect to the dispersive and stochastic orders). The use of truncated distribution is motivated by the fact that the distribution of financial parameters have generally bounded support. Let us first recall the conditions under which some particular distribution functions are ordered with respect to the dispersive order. We refer to [13] for other classes of distribution functions.
Proposition 3.1. Let X and Y be two random variables.
(1) If X ∼ U[a, b] and Y ∼ U[c, d], then X is smaller than Y for the
dispersive order (X ≤disp Y ) if and only if
b− a ≤ d − c.
(2) If X ∼ E(µ) and Y ∼ E(λ), then X is smaller than Y for the
dispersive order (X ≤disp Y ) if and only if
(3) If X ∼ N(m1, σ2) and Y ∼ N(m2, ν2), then X is smaller than Y for
the dispersive order (X ≤disp Y ) if and only if
σ ≤ ν.
As mentioned above, most of the numerical illustrations will be based on model parameter with truncated distribution functions. We present some properties of such distributions.
Definition 3.2. Let X be a random variable with density function f and
(a, b)∈ R2. If F denotes the cumulative distribution of X then the truncated
distribution of X on the interval [a, b] is the conditional distribution of X given that a < X ≤ b. The truncated density function of X is then given by (3.1) f (x|a < X ≤ b) = g(x)
F (b)− F (a)
where g(x) = f (x) for all x such that a < x≤ b and g(x) = 0 else.
In what follows, we denote by NT and ET the truncated normal and the
truncated exponential laws respectively.
Proposition 3.3. Let X and Y be two random variables
(1) if X ∼ NT(m, σ2) where X is truncated on [a, b] then the quantile
function of X is given by
FX−1(x) = φ−1(φ(α) + x(φ(β)− φ(α)))σ + m
where α = a−mσ , β = b−mσ and where φ is the standard normal cumulative distribution function.
(2) if Y ∼ ET(λ) is truncated on [a, b] then the quantile function of Y is
given by
FY−1(x) =−1 λlog(e
−λa+ x(e−λb− e−λa))
where λ denotes the parameter of the exponential distribution.
In the following lemma, we give some conditions that ensure the ordering of two truncated random variables with respect to the dispersive order. Lemma 3.4. Let X and Y be two random variables.
(1) If X ∼ U[a, b] and Y ∼ NT(m, σ2) where Y is truncated on [c, d],
then X is smaller than Y for the dispersive order X ≤disp Y if and only if
b− a ≤ σ√2π(φ(β)− φ(α))
where φ represents the cumulative distribution of a standard gaussian law and where α and β are given by α = c−mσ and β = d−mσ .
(2) If X ∼ ET(µ) and Y ∼ ET(λ) are truncated on the same interval
then X ≤disp Y if and only if
λ≤ µ.
Proof. The previous conditions can be easily derived by differentiating the
difference of the quantile functions FY−1 − FX−1 and by using the fact that
Also, we recall (see Remark 2.2) that if two random variables ordered for the dispersive order have the same finite left point of their support, then they are ordered for the stochastic order. In the example that we will con-sider, NT (resp. ET) denotes the truncated gaussian (resp. exponential)
distribution on [0, 2].
3.1. Value at Risk sensitivity analysis. In risk management, the Value-at-risk (VaR) is a widely used risk measure of the risk of losses associated with portfolio of financial assets (such as stock, bond, etc). From a math-ematical point of view, if L denotes the loss associated with a portfolio of assets, then V aRα(L) is defined as the α-quantile level of this loss, i.e.,
V aRα(L) := inf{x ∈ R : FL(x)≥ α}
where FL denotes the cumulative distribution function of L.
Let us consider a portfolio loss of the form L = ST − K where K is positive
and where ST stands for the aggregate value at time T of a basket of financial
assets. This corresponds to the loss at time T of a short position on this portfolio when the latter has been sold at time 0 for the price K. We assume that S follows a geometric brownian motion so that its value at time T can be expressed as
ST = S0exp (µT + σWT)
where WT is the value at time T of a standard Brownian motion, µ (resp.
σ) is a positive drift (resp. volatility) parameter. Therefore, the α-Value-at-Risk associated with loss L is given by
(3.2) V aRα(L) = S0exp
µT + σ√T φ−1(α)− K.
where S0 ∈ R∗+and where φ−1 is the normal inverse cumulative distribution
function of a standard gaussian random variable. Note that, as soon as α ≥ 0.5, the VaR expression (3.2) can be seen as a product of log-convex non-decreasing functions with respect to µ and σ.
Our interest is to quantify the sensitivity of the uncertain parameters µ and σ on the Value-at-Risk (V aRα(L)) by evaluating the total Sobol indices STµ
and STσ as defined by (2.4). We then analyze how an increase of uncertainty
in the input parameters impacts STµand STσ. In the numerical illustrations,
we consider a VaR associated with a risk level α = 0.9 and for a portfolio loss with the following characteristics:
T = 1, S0= 100, K = 100.
Table 1 illustrates the consistency of total Sobol indices when the distribu-tions of input parameters are ordered with respect to the dispersive order.
Each line of Table 1 corresponds to a scenario where one of the parameter has been increased with respect to both the dispersive and the stochastic order. As can be seen, changing the laws of model parameters µ and σ have a significant impact on the values of total Sobol indices STµ and STσ.
µ∗ µ σ∗ σ ST∗µ STµ S ∗
Tσ STσ
U[0, 1] U[0, 1] U[0, 1] U[0, 2] 0.41 0.20 0.64 0.87 U[0, 2] U[0, 2] U[0, 1] NT(0.5, 2) 0.73 0.48 0.36 0.69
U[0, 1] U[0, 1] ET(5) ET(1) 0.53 0.4 0.52 0.66
U[0, 1] NT(0.5, 2) U[0, 1] U[0, 1] 0.40 0.73 0.65 0.35
Table 1. Total Sobol indices of VaR (3.2) when α = 0.9. All digits are significant with a 95% probability.
3.2. Vasicek model. In risk management, present values of financial or in-surance products are computed by discounting future cash-flows. In market practice, discounting is done by using the current yield curve, which gives the offered interest rate as a function of the maturity (time to expiration) for a given type of debt contract.
In the Vasicek model, the yield curve is given as an output of an instanta-neous spot rate model with the following risk-neutral dynamics
(3.3) drt= a(b− rt)dt + σdWt
where a, b and σ are positive constants and where W is a standard brownian motion. Parameter σ is the volatility of the short rate process, b corresponds to the long-term mean-reversion level whereas a is the speed of convergence of the short rate process r towards level b. The price at time t of a zero coupon bond with maturity T in such a model is given by (see, e.g., [4]): (3.4) P (t, T ) = A(t, T )e−rtB(t,T ) where A(t, T ) = exp (b− σ 2 2a2)(B(t, T )− (T − t)) − σ2 4aB 2(t, T ) and B(t, T ) = 1− e −a(T −t) a .
The yield-curve can be obtained as a deterministic transformation of zero-coupon bond prices at different maturities.
In what follows, we quantify the relative importance of the input parame-ters{a, b, σ} affecting the uncertainty in the bond price at time t = 0. In the following numerical experiments, the maturity T and the initial spot rate r0
are chosen such that T = 1 and r0 = 10%. Table 2 and 3 reports the total
parameter law total index parameter law total index
a U[0, 1] 0.41 a U[0, 1] 0.48
b∗ U[0, 1] 0.52 b U[0, 2] 0.57
σ U[0, 1] 0.18 σ U[0, 1] 0.06
Table 2. Total Sobol indices as a result of a risk perturba-tion of b. All digits are significant with a 95% probability.
Table 3 displays the total Sobol indices when σ∗
≤disp σ and σ∗ ≤st σ.
The law of σ is taken as a truncated gaussian random variable on [0, 2] and has a variance of 0.3. We observe an increase in the total Sobol index of σ and a decrease in total index of a and b.
parameter law total index parameter law total index
a U[0, 1] 0.41 a U[0, 1] 0.25
b U[0, 1] 0.52 b U[0, 1] 0.13
σ∗ U[0, 1] 0.18 σ
NT(0.5, 2) 0.70 Table 3. Total Sobol indices as a result of a risk perturba-tion of σ. All digits are significant with a 95% probability.
3.3. Heston model. In finance, the Heston model is a mathematical model which assumes that the stock price St has a stochastic volatility σt that
follows a CIR process. The model is represented by the following bivariate system of stochastic differential equations (SDEs) (see, e.g., [9])
dSt = (r− q)Stdt +√σtStdBt
(3.5)
dσt = κ(θ− σt)dt + σ√σtdWt
(3.6)
where dhB, W it= ρdt.
The model parameters are • r: the risk-free rate, • q: the dividend rate,
• κ > 0: the mean reversion speed of the volatility, • θ > 0: the mean reversion level of the volatility, • σ > 0: the volatility of the volatility,
• σ0 > 0: the initial level of volatility,
• ρ ∈ [−1, 1]: the correlation between the two Brownian motions B and W .
The numerical computation of European option prices under this model can be done by using the fast Fourier transform approach developed in [6] which is applicable when the characteristic function of the logarithm of Stis known
in a closed form. In this framework, the price at time t of a European call option with strike K and time to maturity T is given by
where for j = 1, 2 Pj = 1 2+ 1 π Z ∞ 0 ℜ e−iφ log Kf j(φ; xt, σt) iφ dφ fj(φ; xt, σt) = exp (Cj(φ; τ ) + Dj(φ; τ )σt+ iφxt) Cj = (r− q)iφτ + κθ σ2 (bj− ρσiφ + dj)τ − 2 log 1 − gj edjτ 1− gj Dj = bj− ρσiφ + dj σ2 1− edjτ 1− gjedjτ gj = bj− ρσiφ + dj bj− ρσiφ − dj dj = q
(ρσiφ− bj)2− σ2(2ujiφ− φ2)
u1 =
1
2, u2 =− 1
2, b1 = κ− ρσ, b2 = κ, xt= log St, τ = T − t. Given that the input parameter are not known with certainty, which one mostly affect the uncertainty of the output pricing function (3.7)? Table 4 displays the total Sobol indices of each parameter under two assumptions on the distribution of input parameter. In the first case (3 first columns), all parameters are assumed to be uniformly distributed. In the second case (3 last columns), we only change the distribution of the interest rate parameter r is such as way that r∗ ≤disp r and r∗ ≤st r. The option characteristics are taken as follows: T = 0.5, S0 = 100, K = 100.
parameter law total index parameter law total index
r∗ U[0, 1] 0.32 r U[0, 2] 0.73 q U[0, 1] 0.39 q U[0, 1] 0.20 κ U[0, 1] 0.0036 κ U[0, 1] 0.0009 θ U[0, 1] 0.0082 θ U[0, 1] 0.0020 σ U[0, 1] 0.0012 σ U[0, 1] 0.0004 σ0 U[0, 1] 0.30 σ0 U[0, 1] 0.08 ρ U[0, 1] 0.0011 ρ U[0, 1] 0.0004 Table 4. Total Sobol indices for the price of a call option in the Heston model. All digits are significant with a 95% probability.
4. Concluding remarks
We have enlightened the fact that Sobol indices are compatible with the stochastic orders theory, and more precisely with the excess wealth order, or the dispersive order, provided that the output function satisfies some mono-tonicity and convexity properties. The Vasicek and Heston model examples suggest that the hypothesis on the form of the output function f might be relaxed (hypothesis of Theorem 2.3 are not fulfilled, especially in the Heston case). In other words, the compatibility between Sobol indices and stochas-tic orders should hold for more general functions than those considered in the present paper. On an other hand, as shown in the examples, the choice of the input laws is crucial: the most influential parameters (in the sense that it corresponds to the greater Sobol index) may change as some input distributions are perturbed. A way to overcome this difficulty could be to use our result on the consistency of Sobol index with stochastic orders to get universal bounds on Sobol indices for a given class of input laws.
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Universit´e de Lyon, Universit´e Lyon 1, Laboratoire SAF EA 2429 E-mail address: areski.cousin@univ-lyon1.fr
Universit´e Paris Sud
E-mail address: alexandre.janon@math.u-psud.fr
Universit´e de Lyon, Universit´e Lyon 1, Institut Camille Jordan ICJ UMR 5208 CNRS
E-mail address: veronique.maume@univ-lyon1.fr
